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Azumaya loci of skein algebras

Hiroaki Karuo, Julien Korinman

Abstract

We compute the Azumaya loci of Kauffman-bracket skein algebras of closed surfaces at odd roots of unity and provide partial results for open surfaces as well. As applications, we give an alternative definition of the projective representations of the Torelli groups derived from non-semisimple TQFTs and we strengthen a result by Frohman-Kania Bartoszynska-Lê about the dimensions of some quotients of the skein modules of closed 3-manifolds.

Azumaya loci of skein algebras

Abstract

We compute the Azumaya loci of Kauffman-bracket skein algebras of closed surfaces at odd roots of unity and provide partial results for open surfaces as well. As applications, we give an alternative definition of the projective representations of the Torelli groups derived from non-semisimple TQFTs and we strengthen a result by Frohman-Kania Bartoszynska-Lê about the dimensions of some quotients of the skein modules of closed 3-manifolds.
Paper Structure (36 sections, 66 theorems, 175 equations, 4 figures)

This paper contains 36 sections, 66 theorems, 175 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background on skein algebras and their representations
  3. Main results
  4. Plan of the paper
  5. Preliminary results on Kauffman-bracket skein algebras
  6. Kauffman-bracket skein algebras
  7. Azumaya locus
  8. Character varieties and deformation quantization
  9. Poisson orders
  10. Witten-Reshetikhin-Turaev representations
  11. Categories of representations of the unrolled quantum groups
  12. Unrolled quantum groups and their representations
  13. The ribbon structure and twisted skein algebra
  14. Renormalized trace and link invariants
  15. Multiplicity modules and trivalent graph calculus
  16. ...and 21 more sections

Key Result

Theorem 1.2

One has $\mathcal{X}^{(1)} \subset \mathcal{AL}$, therefore the Azumaya locus of $\mathcal{S}_A(\Sigma_g)$ is the locus of non-central representations.

Figures (4)

  • Figure 1: On the top: a ribbon graph $G\subset S^3$ and a $\sigma$ decoration. On the bottom: two illustrations of the equality $\left<G, c +\frac{N}{k}c_0 \right>= \left< G, c\right> e^{2i\pi \omega_{\overline{c}}(\widehat{c_0})}$ when $c_0= n e_1+ ne_2$ is a cycle so $\widehat{c_0}=c_0$ and when $c_0=n e_3$ (in this case $\widehat{c_0}$ is contractible).
  • Figure 2: On the left: a trivalent oriented graph $G$, an enumeration $e_1,\ldots, e_{3g-3}$ of its edges and some curves $\beta_1, \ldots, \beta_g$. On the right: two copies $G$ and $G'$ of the graph embedded in $\mathbb{S}^3$ which define a Heegaard splitting of the sphere.
  • Figure 3: Two uni-trivalent graphs of genus $g$ with $n$ external edges.
  • Figure 4: Three curves $\alpha_1, \alpha_2, \alpha_3$ in the holed torus $\Sigma_{1,1}$ and a triangulation $\Delta$.

Theorems & Definitions (134)

  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4
  • ...and 124 more